What is recorded here?
An unruly list of complex differentiable functions can contain a well-behaved subsequence. Montel’s 1907 compactness criterion extracts a locally uniformly convergent subsequence from a locally uniformly bounded holomorphic family. His 1912 work adds a different route: a holomorphic family that omits two shared distinct finite values is normal, allowing a subsequence to diverge uniformly to infinity. These criteria turn questions about many functions into questions about one limiting function.
Source-linked catalogue note. A detailed explanatory guide has not yet been prepared for this entry.
Scientific scope
Convergence is local uniform convergence inside a complex domain, not uniform convergence up to every boundary or convergence of the whole original sequence. The omitted values must be shared and distinct; allowing infinity as a subsequential limit is essential. For meromorphic functions the convergence definition must be changed and a three-value condition is different; it is not asserted here from the holomorphic statement. The 1907 bounded proof treats simple connectivity first and then finite-connectivity/covering extension. The 1912 theorem is recorded in its holomorphic setting; modular-function/logarithm lifting must be applied on suitable simply connected subdomains, not assumed globally single-valued on every multiply connected domain. Earlier Arzela/Ascoli, Weierstrass, Stieltjes/Vitali and Picard/Borel/Schottky/Landau inputs remain distinct.
Read the original work
- Sur les suites infinies de fonctions (1907 original)
- Sur les familles de fonctions analytiques qui admettent des valeurs exceptionnelles dans un domaine (1912 original)
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People and systems
Paul Montel
Source-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.
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English catalogue narrative
Montel: compactness and omitted-value criteria for holomorphic families
An unruly list of complex differentiable functions can contain a well-behaved subsequence. Montel’s 1907 compactness criterion extracts a locally uniformly convergent subsequence from a locally uniformly bounded holomorphic family. His 1912 work adds a different route: a holomorphic family that omits two shared distinct finite values is normal, allowing a subsequence to diverge uniformly to infinity. These criteria turn questions about many functions into questions about one limiting function.
Scientific scope
Convergence is local uniform convergence inside a complex domain, not uniform convergence up to every boundary or convergence of the whole original sequence. The omitted values must be shared and distinct; allowing infinity as a subsequential limit is essential. For meromorphic functions the convergence definition must be changed and a three-value condition is different; it is not asserted here from the holomorphic statement. The 1907 bounded proof treats simple connectivity first and then finite-connectivity/covering extension. The 1912 theorem is recorded in its holomorphic setting; modular-function/logarithm lifting must be applied on suitable simply connected subdomains, not assumed globally single-valued on every multiply connected domain. Earlier Arzela/Ascoli, Weierstrass, Stieltjes/Vitali and Picard/Borel/Schottky/Landau inputs remain distinct.
Reading status
This is a source-linked WRS catalogue note. A full explanatory guide and independent scientific reconstruction have not been completed for this entry. Read the identified original work for its complete narrative and arguments.
People and systems
Paul Montel
Source-listed authors for the exact contribution; experimental subjects are not autonomous scientific authors. Source support and predecessor work retain separate roles. Individual effort shares and unresolved identities are not inferred.